176 MATHEMATICS
Remark : If ABC is a triangle, then by Theorem 10.5, there is a unique circle passing
through the three vertices A, B and C of the triangle. This circle is called the
circumcircle of the ✂ ABC. Its centre and radius are called respectively the
circumcentre and the circumradius of the triangle.
Example 1 : Given an arc of a circle, complete the circle.
Solution : Let arc PQ of a circle be given. We have
to complete the circle, which means that we have to
find its centre and radius. Take a point R on the arc.
Join PR and RQ. Use the construction that has been
used in proving Theorem 10.5, to find the centre and
radius.
Taking the centre and the radius so obtained, we
can complete the circle (see Fig. 10.20).
EXERCISE 10.3
- Draw different pairs of circles. How many points does each pair have in common?
What is the maximum number of common points? - Suppose you are given a circle. Give a construction to find its centre.
- If two circles intersect at two points, prove that their centres lie on the perpendicular
bisector of the common chord.
10.6 Equal Chords and their Distances from the Centre
Let AB be a line and P be a point. Since there are
infinite numbers of points on a line, if you join these
points to P, you will get infinitely many line segments
PL 1 , PL 2 , PM, PL 3 , PL 4 , etc. Which of these is the
distance of AB from P? You may think a while and
get the answer. Out of these line segments, the
perpendicular from P to AB, namely PM in Fig. 10.21,
will be the least. In Mathematics, we define this least
length PM to be the distance of AB from P. So you
may say that:
The length of the perpendicular from a point to a line is the distance of the
line from the point.
Note that if the point lies on the line, the distance of the line from the point is zero.Fig. 10.20Fig. 10.21